Even and odd

I can find out whether a number is even or odd by pairing it off.

Even and odd are the two outcomes of a test anyone can run. The test never asks how big the number is, only whether somebody is left without a partner.

The child gives each verdict with its reason — five is odd, one is left alone — rather than the verdict on its own.

First lesson. Write 5 and 6 on the board and ask which is even. Take the answers, then ask how they would prove it to somebody who said the opposite.

The book states the whole idea in two sentences: even numbers can be grouped by twos, odd numbers cannot.

Children build each of the book's ten sets in counters and run the pair test on it — six jacks, five jacks, four red cars, three oranges, eight triangles — saying the verdict with its reason every time.

A two-column chart headed even and odd, with every number entered only after its pair test has been run.

Children take any odd set from the round, put a finger on the single leftover counter, and say the book's own sentence for it: this number cannot be grouped by twos.

I can seat a group two to a bench and read the verdict off the seating.

The test does not care what is being paired. Seated two to a bench, an odd group always leaves exactly one rider alone, and two left over is impossible because two would take a bench.

The child answers both questions after each seating — how many benches are full, and is anybody sitting with an empty seat beside them — before giving the verdict.

Yesterday the pairs were counters on a desk. Today they are riders on benches, and the leftover is a child with an empty seat beside her.

The book runs one test four times and never names it: everybody pairs up, and the group is even if nobody is left over and odd if exactly one rider is.

Children seat three counters on the bench strip one rider per seat, filling a bench before starting the next, then clear and seat four, seven and ten, answering the two questions aloud after each seating and giving the verdict in the book's own frame.

Four seatings run and recorded in the two columns, each entered only after its seating.

Children look back at the odd seatings, seat five and then nine to check that exactly one rider is alone every time, and say why two riders left over could not happen.

I can show even and odd as two equal rows.

Grouped by twos and two equal rows are one test in two positions. Even numbers make a clean rectangle; odd numbers always bulge, and the bulge is the leftover.

The child's drawings carry the verdict in their shape, so that a stranger reading the page gets the answers from the pictures before the labels.

The pair test has been run standing up, two to a bench. Today the pairs lie down, and the answer becomes a shape.

The pair test has a second face: try to arrange the group as two equal rows.

Children build six as two rows of three matched one above the other, then build five the same way and watch the bulge appear, giving five one honest chance at two equal rows before saying precisely that it cannot.

Three labelled two-row drawings, at least one even and one odd, with the bulge drawn where it happens.

Pages go face up and partners read each other's drawings verdict first, saying what the shape shows before checking the label; a drawing whose shape and label disagree goes back to the counters.

I can draw a group two to a bench so somebody else can read it.

A picture is only a record if it works without the person who drew it. The shaded empty seat is what makes the verdict readable cold.

Another pair reads the drawing back — how many riders, how many full benches, odd or even — without asking anything.

Your two-row drawings were read by a partner who had just built the same thing. Today the reader has not seen your counters at all.

The book draws the idea before it names it: Beth takes the front bench, Millie slides in beside her, there is no room for a third, and Ruby ends up on the bench behind.

On bench paper ruled in twos children draw the book's four groups — three riders, then four, seven and ten — one rider per box, filling a bench before starting the next and shading any box left empty, with the count written underneath.

Four drawings with shaded empty seats and written counts, read back cold by another pair.

Children put the odd drawings together and the even drawings together, say in one sentence what every odd drawing has that no even drawing has, and mark that feature on each one.

I can say what happens to even and odd when one more arrives.

One more always changes the verdict, because the newcomer either partners the leftover or becomes the leftover. That is a claim about every number, not a fact about a few.

The child records the alternating row and can act out both newcomer cases with the counters moving.

Every group so far has sat still while it was tested. Today one counter at a time joins, and the answer will not sit still with it.

Still-life groups get classified in the book; this lesson sets the groups in motion, starting from the book's four apples.

Children build four, run the pair test, then add one counter at a time to nine — predicting the verdict with a reason before each test and recording the verdict letter in a row: E, O, E, O, E, O.

A verdict strip carrying the alternating row, with the always sentence written underneath it.

Children act the two cases with counters — when the group is odd the newcomer partners the leftover, when it is even the newcomer becomes the leftover — and rehearse saying both while the counters move.

I can say whether two joined groups will be even or odd before I count them.

Two odd groups come out even because the two lonely riders find each other. The rule is a prediction, which means it has to be said before the counters move.

The child states the three rules with a build behind each that a partner can check, and marks which one the class settled for itself.

Adding one flipped the answer. Today whole groups are added to whole groups, and the rule has to cover both.

Ruby works out the rule on the spot: her three plus Rob's one has to come out even, and even means nobody rides alone.

Children build three on one strip and one on the other, predict aloud what the joined row will do, slide the strips together and pair the row off, then run the book's other two joins the same way — prediction first, outcome after.

Three of the book's joins predicted, built and reported, and two even groups settled from the counters where the book says nothing.

Children state the three rules they can now defend, each with a build behind it, and mark which two Ruby says out loud and which one is theirs, found on the desk.

I can mark the odd numbers on a path and say why the bare squares are bare.

The odd numbers are not a scattered list. They sit at a fixed distance from each other, and the squares between them are all the same kind of number.

The child marks the path to forty-nine, reads it up and back, and can say why no marker sits on fifty using a bare square as the reason.

The class can test any single number now. Today all the odd numbers get put in one place so the pattern between them can be seen.

Counting by twos is stated and then shown: every other number gets a page and the numbers in between never appear.

Children put a marker on 1, 3, 5, 7 and 9 saying each number as it lands, run a finger between markers to count the squares crossed, then keep the same jump until the path runs out of numbers the book counts.

A marked path to forty-nine, read up and back, with the bare squares named as a group.

On their own paper path children circle every number the book counts to twenty-five, draw an arrow from each circle to the next, write the size of the jump above one arrow and check every other arrow is the same length.

I can say whether a number is even or odd quickly, and prove any call I am challenged on.

Fluency is a call with a proof behind it. The counters stay on the desk as the referee, and every fourth call gets built.

The child calls the sides to twenty at speed and proves every fourth call by pairing, and can tell the two questions in the last round apart.

Seven lessons of testing. Today the calls come fast, and the counters are on the desk for anything anybody argues about.

Kirby's rule turns every number into a side: evens to Emma, odds to Kirby, and the book walks that rule out loud over the numbers one to twenty.

Number cards come up one at a time and children call the side as fast as they can read the card; on every fourth card one pair builds that many counters and pushes them into twos to prove the call, and the class waits for the proof.

The full run to twenty called, with a built proof on every fourth card.

The teacher says one number twice — first as one cow's number, so the child calls the mat it goes to, then as a whole pasture count, so the child deals that many counters and says what lands on each mat — for seven and for twelve, and says which answer is one cow and which is two counts.